{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:NYHW4JSTTUHTQKQUXUIYFJGLLT","short_pith_number":"pith:NYHW4JST","schema_version":"1.0","canonical_sha256":"6e0f6e26539d0f382a14bd1182a4cb5cf5a1aefbfe22a1c7947c956843e0befa","source":{"kind":"arxiv","id":"1209.1158","version":1},"attestation_state":"computed","paper":{"title":"Extending finite group actions on surfaces over $S^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Bruno Zimmermann, Chao Wang, Shicheng Wang, Yimu Zhang","submitted_at":"2012-09-06T02:21:37Z","abstract_excerpt":"Let $OE_g$ (resp. $CE_g$ and $AE_g$) and resp. $OE^o_g$ be the maximum order of finite (resp. cyclic and abelian) groups $G$ acting on the closed orientable surfaces $\\Sigma_g$ which extend over $(S^3, \\Sigma_g)$ among all embeddings $\\Sigma_g\\to S^3$ and resp. unknotted embeddings $\\Sigma_g\\to S^3$.\n  It is known that $OE^o_g\\le 12(g-1)$, and we show that $12(g-1)$ is reached for an unknotted embedding $\\Sigma_g \\to S^3$ if and only if $g = 2$, 3, 4, 5, 6, 9, 11, 17, 25, 97, 121, 241, 601. Moreover $AE_g$ is $2g+2$; and $CE_g$ is $2g+2$ for even $g$, and $2g-2$ for odd $g$.\n  Efforts are made"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1209.1158","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-09-06T02:21:37Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"80df1af3c47cde59205e3d51a71670fa8ff61e2d27cb983c3848708b08b4a163","abstract_canon_sha256":"6ce5e9ca054c27e9415d278f52ea9fd0196fb73fdc238e72ad0c610eba18eb6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:06.376532Z","signature_b64":"RBGgNwUxGZMCY+6oD0JmO/VaZwJjGoyfu7FIstS0Ov/Rjjzm4KAGgI9lUYMr5Mi+scoY9eO2PqM0JjzdXhVXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e0f6e26539d0f382a14bd1182a4cb5cf5a1aefbfe22a1c7947c956843e0befa","last_reissued_at":"2026-05-18T03:46:06.375888Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:06.375888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extending finite group actions on surfaces over $S^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Bruno Zimmermann, Chao Wang, Shicheng Wang, Yimu Zhang","submitted_at":"2012-09-06T02:21:37Z","abstract_excerpt":"Let $OE_g$ (resp. $CE_g$ and $AE_g$) and resp. $OE^o_g$ be the maximum order of finite (resp. cyclic and abelian) groups $G$ acting on the closed orientable surfaces $\\Sigma_g$ which extend over $(S^3, \\Sigma_g)$ among all embeddings $\\Sigma_g\\to S^3$ and resp. unknotted embeddings $\\Sigma_g\\to S^3$.\n  It is known that $OE^o_g\\le 12(g-1)$, and we show that $12(g-1)$ is reached for an unknotted embedding $\\Sigma_g \\to S^3$ if and only if $g = 2$, 3, 4, 5, 6, 9, 11, 17, 25, 97, 121, 241, 601. Moreover $AE_g$ is $2g+2$; and $CE_g$ is $2g+2$ for even $g$, and $2g-2$ for odd $g$.\n  Efforts are made"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1158","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1209.1158","created_at":"2026-05-18T03:46:06.375994+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.1158v1","created_at":"2026-05-18T03:46:06.375994+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.1158","created_at":"2026-05-18T03:46:06.375994+00:00"},{"alias_kind":"pith_short_12","alias_value":"NYHW4JSTTUHT","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"NYHW4JSTTUHTQKQU","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"NYHW4JST","created_at":"2026-05-18T12:27:16.716162+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT","json":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT.json","graph_json":"https://pith.science/api/pith-number/NYHW4JSTTUHTQKQUXUIYFJGLLT/graph.json","events_json":"https://pith.science/api/pith-number/NYHW4JSTTUHTQKQUXUIYFJGLLT/events.json","paper":"https://pith.science/paper/NYHW4JST"},"agent_actions":{"view_html":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT","download_json":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT.json","view_paper":"https://pith.science/paper/NYHW4JST","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.1158&json=true","fetch_graph":"https://pith.science/api/pith-number/NYHW4JSTTUHTQKQUXUIYFJGLLT/graph.json","fetch_events":"https://pith.science/api/pith-number/NYHW4JSTTUHTQKQUXUIYFJGLLT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT/action/storage_attestation","attest_author":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT/action/author_attestation","sign_citation":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT/action/citation_signature","submit_replication":"https://pith.science/pith/NYHW4JSTTUHTQKQUXUIYFJGLLT/action/replication_record"}},"created_at":"2026-05-18T03:46:06.375994+00:00","updated_at":"2026-05-18T03:46:06.375994+00:00"}